Can general relativity be explained by equations describing a fabric of space embedded in a flat 5-dimensional Minkowski space? Does such a set of equations exist or does our universe have an intrinsic curvature that can't be explained by an embedding in a flat Minkowski space of 1 higher dimension? Even if general relativity can be explained by such equations, it still doesn't prove our universe is embedded in a flat Minkowski space of higher dimension.
 A: General relativity in four dimensions does not need to be embedded in a larger space of any sort. Curvature in general relativity is completely defined according to curvatures that are intrinsic induced by parallel translation of vectors. One does not need to have the spacetime in four dimensions embedded in some higher dimension spacetime.
There is general relativity in five dimensions with the flat metric
$$
ds^2~=~dt^2~\pm~du^2~-~dx^2~-~dy^2~-~dz^2.
$$
The constraint
$$
t^2~\pm~u^2~-~x^2~-~y^2~-~z^2~=~\alpha^2
$$
defines hyperboloids embedded in this flat spacetime. The condition $\pm u^2$ defines the anti-deSitter $(+du^2)$ and de Sitter spacetimes $(-du^2)$. The constant $\alpha$ defines the cosmological constant. So this is a special case of what you are talking about.
A: For Riemannian manifolds, I believe the best result currently known is that a manifold of dimension $n$ can be isometrically embedded in a euclidean space of dimension $2(2n+1)(3n+7)$.  So, for example, a 3-dimensional spacelike slice of spacetime can be embedded in a flat euclidean space of at most 224 dimensions.  Maybe in low-dimensional cases like this one can do better, but if so I'm not aware of it.
So much for space.  If you want to embed all of spacetime, I think the best known result is that every Lorentzian manifold can indeed be embedded in a flat Lorentzian manifold, but I don't think any bound is known on the necessary number of dimensions.
Edited to add:  I see by the reference I posted in the comments that there is in fact a known bound for Lorentzian manifolds:  $2(2n+1)(2n+6)$.  So you can imbed all of (four dimensional) spacetime in a copy of $R^{252}$, with signature $(126,126)$
A: This is an afternote to WillO's answer which cites:
Robert E. Greene, "Isometric Embeddings," Bull. AMS 1969
which addressed known bounds on the dimension required of flat Euclidean / Minkowsian space if it is to be an embedding for a solution of the Einstein field equations, which of course is a four-dimensional signatured manifold.
It's worth noting that important special cases one can be embedded in much lower dimensions than the insanely loose bounds defined by the Nash embedding theorem and its Lorentzian equivalents. Such simplifications happen in cases of high symmetry. For example, the large scale homogeneous/ isotropic universe defined by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric can indeed be thought of as being embedded in five dimensional space, with signature $(1,\,4)$, because it can be partitioned into foliations of space, with the foliations indexed by a universal time co-ordinate, and the spatial foliations are isometric to three dimensional spheres / hyperboloids in $\mathbb{R}^4$ with only the scale factor and energy / pressure evolving over time.
Indeed, the lecture notes:
Balša Terzić, Lecture notes for PHYS 652, Old Dominion University
take the unusual approach of lifting well known  19th century geometry results on the hypersphere / hyperboloid and linking them to a homogeneous, isotropic stress energy tensor. The Ricci tensor is of course diagonal in this case, so the student is relieved of the full complexity of GR and gets to see the direct link between stress energy (in the diagonal case) and curved spaces. 

Lawrence B. Crowell's answer cites two other examples very like FRLW which can be split into spatial foliations such that the whole spacetime is embedded in 5 dimensions with a $(1,\,4)$ signature - the de Sitter and anti-de Sitter spaces, which are like FLRW but with no matter and pressure (vacuum solution) and a special value for the cosmological constant.
A: To start with, a manifold is not always able to be embed in higher dimension, especially when singularity (black hole) involves. 
I would more agree if it is described by a 3-d gravity-free field theory. This is similar to the idea named AdS/CFT duality. Of course here is not AdS space, but the spirit is similar, I think. 
But I'm not an expert in this, so... 
A: Maybe our universe is a uniformly accelerating fabric embedded in a 5-dimensional Minkowski space and doesn't actually obey the laws of general relativity. If gravity is entirely the result of objects making a dent in it with their weight, then for objects with an escape velocity much slower than the speed of light like Earth, the gravitational time dilation is exactly the amount general relativity predicts. The emission of a gravitional wave from merging black holes which we detected is also consistent with this theory of gravity. If that really is how our universe works, an object in free fall will still follow a geodesic of the fabric of space but there will be no gravitomagnetism and an object of sufficiently low mass if it's compressed dense enough might form a singularity then disappear at it and then the singularity will rush towards the rest of the fabric of space faster than light and cease to exist once its speed goes down to the speed of light, destroying matter and its gravitational field. Maybe at the quantum level, the information of what went into the singularity is preserved as disordered vacuum fluctuations outside the fabric of space. How do we know our universe follows the laws of general relativity if we haven't made detailed enough measurements of the behaviour of objects in a gravitational field or observed gravitomagnetism? Why should general relativity be true? Just because angular momentum has been proven to be conserved according to simplified laws of physics doesn't mean it's also conserved in the formation of a black hole. The gravitational constant is determined by the acceleration of the fabric and its resistance to bending but we already know what the gravitational constant is. Given what the gravitation constant is, the faster the fabric is accelerating, the lower the minimum mass that can be compressed to form a permanent black hole. Although Einstein's field equations are probably the simplest possible equations that are consistent with observations and are preserved at any point at any velocity below c at any orientation and are consistent with observations, there's no reason to be sure the universe isn't an embedded fabric whose equations do not simplify to such simple equations that described only the space itself According to https://en.wikipedia.org/wiki/Binary_star#Cataclysmic_variables_and_X-ray_binaries, one member of a binary system is believed to be a black hole. It's probably because of detection of movement of the other stat in the system that it's believed that there's a black hole in the system. If our universe does work the way I described, the existence of a black hole of that mass rules out the possibility that the fabric is accelerating below a certain acceleration.
